Have you tried a soft cloth dipped in warm soapy water then well wrung out and gently rub the mark on the screen. Make sure monitor is turned off and if its one with the power pack inside, allow the screen to cool. [They do get warm enough to cause smearing when cleaning].
The above will not cause any harm at all. Anything else is at your own risk and depends on how bad the mark is, and how much its bothering you.
I have been told that its perfectly safe to use pure alcohol. In fact some of the dedicated screen cleaners use a percentage of this in their content.
Last suggestion is one that I would try only if nothing else works, but I can't see it doing harm. [CAUTION FROM HERE ON] and that is to use a little WD40 sprayed onto a soft cloth. It is an excellent product for removing stains such as ink or marker from most surfaces without causing damage.
One use I've found for it is to remove the sticky residue left on glass or plastic after removing a label or price tag. It really does remove the residue with no effort and easy to then wash off in soapy water. I have used this myself on many delicate articles with excellent results.
If you do try this last method, then once you've cleaned off the ink mark, remove the film of oil with a soft cloth wrung out in warm soapy water. Clean the whole of the screen down with this cloth, then buff to a high sheen with a soft dry cloth.
I can't guarantee the outcome of the latter method, so its down to you to take the risk, but I can't see it making matters worse than what they already are.